High School How to tell if a quantity is defined as a derivative or a ratio

[etotheipi]

I appreciate that this is perhaps a strange question but it's been bugging me a little.

For instance, velocity is defined as the time derivative of position, so will always appear as the gradient a graph of x vs t. However, something like resistance as R = V/I is defined in terms of a ratio, and it would not be correct to interpret it as the gradient unless in the special case where current happens to be constant. Instead, we would determine R graphically by reading off values.

Is there an easy way to tell which is true just by looking at the formula or must they just be learned? One rule I've sort of made up is that when the equation includes time in some form, e.g. Q = It, it is likely actually defined in terms of a derivative like dQ/dt, since we're dealing with a rate of change. All other equations I can think of that don't include time seem to be defined with ratios, for instance things like V=IR, M=pV, F=PA etc.

This seems to work pretty well, but I don't want to learn something wrong that might cause confusion in the future. Can you think of any exceptions to this?

Edit

I just remembered dW/dx = F as a counter example, for instance if we plotted the cumulative work done as a function of displacement. Evidently my rule is a bit wrong.

 

[anorlunda]

However, something like resistance as R = V/I is defined in terms of a ratio, and it would not be correct to interpret it as the gradient unless in the special case where current happens to be constant. Instead, we would determine R graphically by reading off values.

Not true. We can and do use R values as the slope of the tangent to the curve. Other times we can use the average R over some region of V and I. R=dV/dI is also useful at times. So R=V/I is only one of several definitions for R.

But in general, any derivative is just the ratio of infinitesimal differences. So I would not put such great importance on ratios versus derivative.

One amusing quote I remember from a teacher. "Differential calculus is just fun and games with subtraction. Integral calculus is just fun and games with addition."

 

[etotheipi]

Not true. We can and do use R values as the slope of the tangent to the curve. Other times we can use the average R over some region of V and I. R=dV/dI is also useful at times. So R=V/I is only one of several definitions for R.

But in general, any derivative is just the ratio of infinitesimal differences. So I would not put such great importance on ratios versus derivative.

One amusing quote I remember from a teacher. "Differential calculus is just fun and games with subtraction. Integral calculus is just fun and games with addition."

Thank you for your reply, I just googled differential resistance and it's pretty interesting.

However, there does seem to be a material difference. If we take a displacement time graph which, for instance, might take the form of a sine wave. If we pick one point on the curve and measure the gradient, we obtain the velocity. If we pick another point with the same gradient, perhaps one cycle later, it will also give the same velocity. Though evidently the quantity x/t will be different for both points.

This causes me a bit of confusion when it comes to things like Ohm's law. The definition of resistance in my spec is the ratio of values read off the graph, though myself and many of my friends wouldn't necessarily know this just given the formula as gradients are so common for physics exams.

If we plot a graph of mass against volume for an object whose density is perhaps varies with volume, we would obtain the values of density by reading off the graph since density is defined as the constant of proportionality, so a gradient method (which might work in the case of constant density) would be wrong.

This is what's bothering me a little

 

[anorlunda]

The definition of resistance in my spec is the ratio of values read off the graph

You don't have to insist that the values go back to the origin.

For a simple single-valued curve, nearly linear it doesn't make much different. But there is no law that forbids a V-I curve that is many valued and almost any nonlinear shape. In a case like that, a line from the present V, I back to the origin is pretty useless. Differential R might be useful. But we could also solve a circuit containing such a device without ever defining any R, we just have V=f(I) or I=f(V).

Here's a silly example. No real-life device I know looks like that, but I'm just illustrating the generality of V versus I relationships.

 

[etotheipi]

You don't have to insist that the values go back to the origin.

For a simple single-valued curve, nearly linear it doesn't make much different. But there is no law that forbids a V-I curve that is many valued and almost any nonlinear shape. In a case like that, a line from the present V, I back to the origin is pretty useless. Differential R might be useful. But we could also solve a circuit containing such a device without ever defining any R, we just have V=f(I) or I=f(V).

Here's a silly example. No real-life device I know looks like that, but I'm just illustrating the generality of V versus I relationships.

View attachment 249963

I guess this makes sense, my definition of resistance is perhaps slightly arbitrary.

But on a completely different note, what if we consider that mass-volume equation. It would be silly to define density as dM/dV, for an object whose density might in this case vary with its volume. There are definitely some cases where an equation contains a constant of proportionality (though not necessarily "constant" per se) where it would be wrong to measure the gradient.